If ` Y= log_7 (log_7 x),` then ` (dy)/(dx)` =

Options

A. `(1)/(log 7 . x log x)`
B. `(log 7 )/( log x)`
C. `(x log x)/(log 7)`
D. None

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( Y = \log_7(\log_7 x) \), we will follow these steps:

### Step 1: Rewrite the logarithm in terms of natural logarithm
Using the change of base formula for logarithms, we can express \( Y \) as:
\[ Y = \frac{\log(\log x)}{\log 7} \]
where \( \log \) denotes the natural logarithm (base \( e \)).

### Step 2: Differentiate \( Y \) with respect to \( x \)
Now, we differentiate \( Y \) using the quotient rule. Since \( \log 7 \) is a constant, we have:
\[ \frac{dy}{dx} = \frac{1}{\log 7} \cdot \frac{d}{dx}[\log(\log x)] \]

### Step 3: Differentiate \( \log(\log x) \)
To differentiate \( \log(\log x) \), we apply the chain rule:
\[ \frac{d}{dx}[\log(\log x)] = \frac{1}{\log x} \cdot \frac{d}{dx}[\log x] \]
Now, we know that:
\[ \frac{d}{dx}[\log x] = \frac{1}{x} \]
Thus, substituting this back, we get:
\[ \frac{d}{dx}[\log(\log x)] = \frac{1}{\log x} \cdot \frac{1}{x} = \frac{1}{x \log x} \]

### Step 4: Substitute back into the derivative
Now substituting this result back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\log 7} \cdot \frac{1}{x \log x} \]

### Step 5: Final expression for the derivative
Thus, we can write the final expression for \( \frac{dy}{dx} \) as:
\[ \frac{dy}{dx} = \frac{1}{x \log x \log 7} \]

### Summary
The derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \frac{1}{x \log x \log 7} \]